Problem: Simplify. $i ^ {9}$
Solution: The most important property of the imaginary unit $i$ is that ${i ^ 2} = {-1}$ When this property is plugged into $i ^ 4$ , we get: $i ^ 4 = ({i ^ 2}) ^ 2 = ({-1}) ^ 2 = 1$ So, we can reduce the exponent by multiples of 4 and have the same result. The remainder after dividing 9 by 4 is 1, so $i ^ {9} = i ^ {1}$ Anything to the first power is the number itself. $i ^ 1 = i$ $i ^ {9} = i ^ {1} = i$.